Pinning synchronization of complex dynamical networks with and without time-varying delay

The pinning synchronization in two types of complex dynamical networks are studied in this paper. In the first one, the nodes are coupled by their states; In the second one, the nodes are coupled by their past states or delayed states. By designing suitable pinning control schemes, several synchronization criteria are derived for these proposed network models. Moreover, some adaptive strategies are developed to get proper coupling strength for the first network model. For the second network model, we give several synchronization criteria by utilizing the designed pinning adaptive feedback controllers. Finally, two numerical examples indicate that complex dynamical networks with and without time-varying delay can achieve synchronization by pinning a small fraction of nodes.